In illustrating the special status of mathematics, I looked for a
theorem of a thematic nature, whose proof is very well known, and
where the appropriate axiomatic treatment of the statement and of the
proof is also very well known. Also, the proof has substantial depth,
both in an absolute sense, and as compared to "proofs" in other fields.
THEOREM. Any polynomial in one real variable with real coefficients,
of degree at most 5, with at least six zeros, is degenerate (all
coefficients are zero).
THEOREM. Let a,b,c,d,e,f,x,y,z,u,v,w be real numbers. Suppose that x <
y < z < u < v < w, and for all real numbers t, if t = x or t = y or t
= z or t = u or t = v or t = w, then at^5 + bt^4 + ct^3 + dt^2 + et +
f = 0. Then a = b = c = d = e = f = 0.
In what subjects outside mathematics do we have comparable
achievements or phenomena?
In normal attempts to give an answer to this, one does confront the
interesting issue of proofs in physical science, where there is
careful reasoning, with a (often very) substantial mathematical
kernel, and that kernel is of the above general kind or generally much
more complicated and deeper. E.g., various kinds of ordinary and
partial differential equations are used, or differential geometry,
etcetera.
However, it would seem that this is a conjoining of a piece of
mathematics and some perceptive reasoning, where the latter is never
cast as a proof in any comparable sense to what we have in mathematics.
There is the rather crucial problem as to whether the "non
mathematical component" can actually be turned into a "proof" in any
reasonable sense. This would require major advances in the foundations
of physical science.
Computer science is another important area to try to analyze in these
terms. I.e., are there proofs in any clear sense in computer science
that are not mathematics? If so, what kind of depth do they exhibit?
This reminds me of a phrase that G. Kreisel used in print and in
conversation: informal rigor. (Perhaps this phrase goes back to
Goedel?) But the examples I remember him citing are extremely
mathematical. He emphasized the example of a rigorous proof that the
informal concept of validity in mathematical structures coincides with
formal provability in a usual setup for predicate calculus. You don't
have to formalize what a mathematical structure is in any usual way -
like tuples of sets of certain kinds. You can instead use evident
properties of the notion.
Of course, this is so heavily mathematical, that it doesn't really say
much about the present FOM interchanges denying and affirming the
special status of mathematics.
THESIS: Any remotely convincing case that mathematics does not have
special status with regard to proof, will involve or precipitate a
major breakthrough in the foundations of subjects other than
mathematics.
CHALLENGE: Show us by specific convincing examples here on the FOM,
not involving pointers to unstructured prose appearing elsewhere.
TECHNICAL PROJECT: Make a detailed study of the structure of proofs of
the above Theorem (also for higher degrees).
Harvey Friedman